Showing papers on "Matrix differential equation published in 1992"
TL;DR: In this article, the stability radius of a matrix pair is defined to be the norm of the smallest perturbation Delta A such that (A+ Delta A, B) is unstable.
Abstract: The concept of stability radius is generalized to matrix pairs. A matrix pair is said to be stable if its generalized eigenvalues are located in the open left half of the complex plane. The stability radius of a matrix pair (A, B) is defined to be the norm of the smallest perturbation Delta A such that (A+ Delta A, B) is unstable. The purpose is to estimate the stability radius of a given matrix pair. Depending on whether the matrices under consideration are complex or real, the problem can be classified into two cases. The complex case is easy and a complete solution is provided. The real case is more difficult, and only a partial solution is given. >
83 citations
82 citations
TL;DR: In this paper, a similarity transformation is used to transform complex eigenvalues and eigenvectors into real eigen values and eignevectors, thereby permitting spectra and profiles of pure constituents to be estimated.
Abstract: Recently several papers have described the generalized rank annihilation method; however, in some cases complex eigenvalues and eigenvectors may appear when the generalized eigenproblem is solved. When complex eigenvalues and eigenvectors are encountered, the results cannot be used to estimate pure component profiles (e.g. spectra or chromatograms). In this paper, a similarity transformation is used to transform complex eigenvalues and eigenvectors into real eigenvalues and eigenvectors, thereby permitting spectra and profiles of pure constituents to be estimated. The modified GRAM method is illustrated with simulated and real data.
64 citations
49 citations
TL;DR: For all n > 3, every matrix in K has at least three distinct eigenvalues; such a matrix has exactly three distinct Eigenvalues if and only if it is a Hadamard tournament matrix as discussed by the authors.
34 citations
TL;DR: In this article, upper bounds for summations including the trace, and for products including the determinant, of the eigenvalues of the solution of the continuous algebraic Lyapunov matrix equation are presented.
Abstract: Upper bounds for summations including the trace, and for products including the determinant, of the eigenvalues of the solution of the continuous algebraic Lyapunov matrix equation are presented. The majority of the bounds are tighter than those in the literature, and some are new. >
29 citations
TL;DR: In this article, a class of second order ordinary differential equations with jumping nonlinearities was considered, and the existence of 2 k solutions was shown by means of a Morse index computation, under zero boundary conditions.
25 citations
TL;DR: The eigenvalues of the ZN invariant model transfer matrix for N odd strings were found in this paper, and their zeros in the complex plane of the rapidity variable were shown to satisfy a set of Bethe-ansatz type equations similar to those obtained for the integrable XXZ chains.
Abstract: The eigenvalues of the Fateev-Zamolodchikov ZN invariant model transfer matrix are found for N odd. Their zeros in the complex plane of the rapidity variable are shown to satisfy a set of Bethe-ansatz type equations similar to those obtained for the integrable XXZ chains. The eigenvalue for a filled sea of (N-1)-strings gives the free energy found by the matrix inversion method.
21 citations
TL;DR: For hermitian pencils it is proved that the Puiseux expansion generates groups of eigenvalues ordered according to the partial algebraic multiplicities of the unperturbed eigenvalue.
Abstract: Perturbation properties of a quadratic matrix pencil containing a “small” parameter are considered. Main results concern the splitting properties of multiple eigenvalues and the corresponding Puiseux expansions. For hermitian pencils it is proved that the Puiseux expansion generates groups of eigenvalues ordered according to the partial algebraic multiplicities of the unperturbed eigenvalue.
19 citations
TL;DR: In this paper, the authors derived an autonomous system of real first-order ordinary differential equations by using SU(3) notation, and performed a Painleve analysis to distinguish integrable and non-integrable cases.
Abstract: From the discrete self-trapping (DST) equation for three degrees of freedom rewritten in terms of density matrix one can derive an autonomous system of real first-order ordinary differential equations by using SU(3) notation. Performing a Painleve analysis, it is found that, depending on the parameters of the system, integrable and non-integrable cases can be distinguished.
16 citations
TL;DR: In this article, model-reduction algorithms for structural dynamics systems were developed for a structural dynamics system described by a second-order matrix differential equation, which can eliminate control and observation spillovers while requiring only the dynamic spillover terms to be considered.
Abstract: Krylov vectors and the concept of parameter matching are combined here to develop model-reduction algorithms for structural dynamics systems. The method is derived for a structural dynamics system described by a second-order matrix differential equation. The reduced models are shown to have a promising application in the control of flexible structures. It can eliminate control and observation spillovers while requiring only the dynamic spillover terms to be considered. A model-order reduction example and a flexible structure control example are provided to show the efficacy of the method.
TL;DR: In this article, a characterization of real stable matrices for which the unstructured complex and real stability radii are equal is presented. But neither the case of equality nor strict inequality is generic.
Abstract: Characterization of those real stable matrices for which the unstructured complex and real stability radii are equal is presented. Special cases of this equality are discussed. As application of this result, it is shown that, for 2*2 matrices with nonreal eigenvalues, neither the case of equality nor the case of strict inequality is generic. >
TL;DR: In this paper, an existence theorem for the eigenvalues of a spectral problem is studied, and it is shown that the original problem is equivalent to that of determining the characteristic values of a linear (non-selfadjoint) compact operator.
Abstract: An existence theorem for the eigenvalues of a spectral problem is studied in this paper. The physical situation behind this mathematical problem is the determination of the eigenfrequencies and eigenmotions of a fluid-solid structure. The liquid part in this structure is represented by a viscous incompressible fluid, while the solid part is a set of parallel rigid tubes. The spectral problem governing this system is a quadratic eigenvalue problem which involves Stokes equations with a non-local boundary condition. The strategy for tackling the question of existence of eigenvalues consists of proving that the original problem is equivalent to that of determining the characteristic values of a linear (non-selfadjoint) compact operator. Sharp estimates for the eigenvalues give precise information about the region of ω where the eigenvalues are located. In particular, we prove that this problem admits a countable set of eigenvalues in which only a finite number of them have a non-zero imaginary part.
TL;DR: The analysis of the errors involved in approximate orthogonalization with respect to previously found eigenvectors in preconditioned iterations of a subspace for simultaneous determination of a cluster of eigenvalues and the corresponding eigenvctors of a large sparse symmetrical eigenvalue problem is presented.
Abstract: We present the analysis of the errors involved in approximate orthogonalization with respect to previously found eigenvectors in preconditioned iterations of a subspace for simultaneous determination of a cluster of eigenvalues and the corresponding eigenvectors of a large sparse symmetrical eigenvalue problem.
TL;DR: In this article, the problem of estimating the eigenvalues of the scale matrix Δ is considered and new class of estimators are obtained which dominate the best linear estimator of the form cF.
Abstract: Let F have the multivariate F distribution with a scale matrix Δ. In this paper, the problem of estimating the eigenvalues of the scale matrix Δ is considered. New class of estimators are obtained which dominate the best linear estimator of the form cF. Simulation study is also carried out to compare the performance of these estimators.
16 Dec 1992
TL;DR: In this article, the exponential nature of the convergence of the solution of the time-invariant matrix Riccati differential equation toward the stabilizing solution of a linear quadratic LQ problem is discussed.
Abstract: The exponential nature of the convergence of the solution of the time-invariant matrix Riccati differential equation toward the stabilizing solution of the algebraic Riccati equation is displayed on an explicit formula. It is assumed that the system is stabilizable and the Hamiltonian matrix has no eigenvalues on the imaginary axis. Computable characteristics are given which can be used to estimate how well a large finite horizon linear-quadratic (LQ) problem is approximated by an infinite horizon LQ problem. >
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TL;DR: In this paper, the authors define universal invariance in the synthetic form of F and show that any invariant function η: FXA → D general is invariant in terms of linear dependencies among the elements (columns) of a family.
Abstract: ing d = f · b. Conversely, if this homomorphic condition holds everywhere, then apply b to both its sides, observe that Cχb = iA and by abstracting c get h = Cχa with a = h · b. Q.D.E. 5.8 Definitions. Keep the notation of 5.4 and let B be the set of bases of α with index X. Consider a family k: B → F(FXA)C, for some C, and a b ∈ B. If for all b′ ∈ B and M :X → A (43) kbM = kb′(M ◦ b′) , then we say that the function kb: FXA → C, as well as family k, are (absolutely) invariant. (In fact, if kb is invariant, then any kb′ is, as it follows from 5.1 through easy passages.) See [38] for a (lone) definition of universal invariance in F. Klein’s synthetic form. We call an invariant function η: FXA → D general if for any invariant family k as above there is a function f : B → FDC, such that kb′ = fb′ · η for all b′ ∈ B. 5.9 Corollary. If χ is an analytic representative, then η = Cχ is a general invariant function. Proof. By 5.6 η = rb−1: FXA → H αα. Hence, η is an invariant function if (43) holds for the k such that ka = ra−1 for all a ∈ B, i.e. if rb′(ηM) = M ◦ b′ for all M :X → A and b′ ∈ B. This is a trivial identity because of our notation of r, η and ◦ in 1.2, 5.9 and 5.4. Also, η is general, since it is one to one. (We can take fa = ka · η−1 for all a ∈ B.) Q.D.E. 5.10 Example. By the preceding corollary any function or predicate of the endomorphism associated to a family M is invariant. Hence, in based algebras all present theory about universal eigenvalue equations concerns invariants only. E.g. by 5.7(A) independence is an invariant predicate. This differs from Universal Algebra, where noninvariant notions are accepted, as we are going to show. In fact, we disprove the invariance of the “C/Ci–independences” as in [15]. These are weaker notions of independence, sometimes [11] and [17] considered akin of independence itself. In a vector space, they express the lack of certain linear dependencies among the elements (columns) of a family M and are equivalent to the independence of M . Here, we recall a couple of them that correspond to conditions (C3) and (10) of [15]. (However, it is easy to see that the next conterexample works even for conditions (C1) and (C2) ibidem.) Given M and α as in 5.8, consider the following two conditions: ⋂ (C↑V ) = C( ⋂ V ) , for all V ⊆ PM (44) and Mx ∈ Cv implies Mx = vx , for all v ⊆ M and x ∈ X , (45)
TL;DR: In this article, a new dynamical system of Lax type is presented to solve the skew-Hermitian eigenvalue problem, which converges to a diagonal matrix which is a permutation of the eigenvalues of the initial value matrix.
Abstract: This paper presents a new dynamical system of Lax type which solves the skew-Hermitian eigenvalue problem. The solution of the system is found to converge to a diagonal matrix which is a permutation of the eigenvalues of the initial value matrix.
TL;DR: In this article, an inequality between the spectral norm of R and the L/sub Omega /-norm of its frequency response is derived, and qr-factorizations and bounds for the spectral norms of such realization matrices are derived.
Abstract: Examines the realization matrix R=(A b; c d) defined by a state variable model of a linear, shift invariant, discrete time, scalar system. Several properties concerning the eigenvalues and singular values are derived, which are used to obtain tests for the minimality of the state variable model. An inequality is derived between the spectral norm of R and the L/sub Omega /-norm of its frequency response. The realization matrices of principal axis realizations are characterized in terms of their eigenvalues and singular values. qr-factorizations and bounds for the spectral norms of such realization matrices are derived. >
TL;DR: In this paper, a homogeneous linear matrix equation related to the block similarity of rectangular matrices is studied and the dimension of the vector space of its solutions is obtained and a characterization of block similarity by rank tests is given.
Abstract: In this paper we study a homogeneous linear matrix equation related to the block similarity of rectangular matrices. We obtain the dimension of the vector space of its solutions and we describe these solutions. We give a characterization of the block similarity by rank tests. We extend Roth's criterion to the corresponding non homogeneous equation.
TL;DR: In this paper, a new method is proposed for obtaining a few eigenvalues and eigenvectors of a large-scale RPA-type equation, which is applicable to the eigenvalue problems of nondiagonally dominant matrices.
Abstract: A new method is proposed for obtaining a few eigenvalues and eigenvectors of a large-scale RPA-type equation. Some numerical tests are carried out to study the convergence behaviors of this method. It is found that the convergence rate is very fast and quite satisfactory. It depends strongly on the way of estimating the deviation vectors. Our proposed scheme gives a better estimation for the deviation vectors than Davidson's scheme. This scheme is applicable to the eigenvalue problems of nondiagonally dominant matrices as well. Keywords: large-scale eigenvalue problem, RPA-type equation, fast convergence.
TL;DR: An application of the compound matrix method for the solving Orr-Sommerfeld equation for high wave numbers is presented in this article, where the problem of evaluating the eigenfunctions become difficult.
TL;DR: Yiantsios and Higgins as discussed by the authors presented an algorithm based on a compound matrix method for solving difficult eigenvalue problems of n equation sets in connected domains that are coupled through (n - 1) sets of interfacial boundary conditions, when n is an arbitrary number.
Abstract: SUMMARY An algorithm based on a compound matrix method is presented for solving difficult eigenvalue problems of n equation sets in connected domains that are coupled through (n - 1) sets of interfacial boundary conditions, when n is an arbitrary number. As an example, a linear stability problem of n-layer plane Poiseuille flow is formulated. The resulting Orr-Sommerfeld equations form a set of stiff differential equations at high wavenumbers, which are solved accurately for various combinations of parameters. and Davey' for two-point boundary value problems and eigenvalue problems of the Orr-Sommerfeld equation. These investigations computed eigenvalues and eigenfunctions with marginal errors where standard shooting methods failed. Yiantsios and Higgins6 extended the method to equation sets valid over two connected domains that are coupled through interfacial conditions; in particular, they solved the Orr-Sommerfeld equations for two superposed fluids in plane Poiseuille flow. However, since the eigenvalues are obtained by matching the interfacial boundary conditions, the method is unsuitable for computations in more than two domains connected by interfacial boundary conditions. Here, the compound matrix method is implemented for n equation sets that are valid over connected domains through (n - 1) sets of interfacial boundary conditions, where n is arbitrary. Instead of matching boundary conditions at a particular interface for finding eigenvalues, the integration of the compound differential system proceeds with new initial conditions at an interface for the compound differential system of the next domain. Subsequently, the eigenvalues are computed by matching the boundary conditions at the end of the last domain. This general algorithm for finding eigenpairs is used in the linear stability analysis of n-layer, Newtonian, plane Poiseuille flow, for which resulting differential systems are known to be stiff at large wavenumbers.
TL;DR: Recently developed approaches to the analysis and solution of the matrix Riccati equation encountered in continuous-time control and filtering problems are adapted for use.
Abstract: The classical state-space approach to continuous-time spectral factorization is used to develop a method for discrete-time spectral factorization. This results in a quadratic matrix equation which is referred to as the modified algebraic Riccati equation (MARE). Solving this equation for the factorizing solution constitutes the major part of the computation needed to obtain the spectral factor. Recently developed approaches to the analysis and solution of the matrix Riccati equation encountered in continuous-time control and filtering problems are adapted for use. >
TL;DR: All problems and solutions should be sent, typewritten in duplicate, with complete address, to Murray S. Klamkin, Department of Mathematics, University of Alberta, Edmonton, Alberta T6G 2G1 Canada.
Abstract: All problems and solutions should be sent, typewritten in duplicate, with complete address, to Murray S. Klamkin, Department of Mathematics, University ofAlberta, Edmonton, Alberta T6G 2G1 Canada. An as’terisk placed beside a problem number indicates that the problem was submitted without solution. Proposers and solvers whose solutions are published will eceive 5 reprints of the correspondingproblem section. Other solvers will receivejust one reprintprovided a self-addressed stamped (U.SM. or Canada) envelope is enclosed. Proposers and solvers desiring acknowledgment oftheir contributions should include a self-addressed stampedpostcard. (No stamps necessaryfor outside the U.Salt. and Canada.) Solutions should be received by June 30, 1992.
TL;DR: By the use of interval arithmetic, the bounds are secured against rounding errors and the eigenvalues are proved rigorously to have multiplicity equal to one.
Abstract: A new procedure is proposed for the calculation of bounds for simple eigenvalues of a real symmetric parameter-dependent matrix. By the use of interval arithmetic, the bounds are secured against rounding errors; thus, the eigenvalues are proved rigorously to have multiplicity equal to one.
TL;DR: New approximation schemes for the nonlinear matrix Riccati differential equation are presented from any one-step or multistep method applied to the original linear quadratic control problem and it is proved that this rate is the same as the one of the initially chosen scheme.
Abstract: The object of this paper is to present new approximation schemes for the nonlinear matrix Riccati differential equation They are obtained from any one-step or multistep method applied to the original linear quadratic control problem They lead to the same type of schemes However only one matrix inversion is required at each discretization node even if the Riccati equation is a nonlinear equation This important computational advantage is obtained without altering the original nodal asymptotic convergence rate It is proved that this rate is the same as the one of the initially chosen scheme